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Visualizing the Parabola

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Visualizing the Parabola

Algebra-2

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You will change the parameters of a quadratic function and see the effects on the graph.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Interpret the effects of the coefficients of a quadratic function on the orientation, shape and location of its graph.
  • Calculate and interpret the discriminant value to find the number of x-intercepts of the parabola.

Everything You'll Have Covered

What are the forms for the equation of a quadratic equation?

The vertex form for a quadratic equation is f(x) = a(x - h)^2 + k,

The general form for a quadratic equation is f(x) = ax^2 + bx + c, where a is different from zero in both forms.

What is the effect of each coefficient on the graph of a parabola?

Orientation:

The orientation (upward or downward) look of a parabola is shown first based on the coefficient "a" and its sign. If "a" is positive then the orientation of the parabola will be going upward (happy face) and if "a" is negative then the orientation will be going downward (sad face). Next, if the value of "a" changes to a larger or smaller number then we see that this affects the width of the parabola. If "a" is larger than one then the parabola will be narrower and if it is less than one then it will be wider.

Location - Vertex Form:

When the "h" value is added to the vertex form of an equation then it affects the horizontal shift (moves the vertex). It is shown how it moves the "opposite" direction of that number. For example, if the equation is y = (x - 2)^2 then this will shift the parabola two units to the right and vice versa.

The effect of "k" is that it shifts the parabola up or down. The amount of the "k" coefficient is the vertical shift of the parabola. For example, if the equation was y = (x^2)- 4 then this parabola would be shifted four units down.

Location - General Form:

In the general form of a quadratic function the effect of the coefficients also make an impact. if c increases, the parabola moves up, and if "c" decreases, the parabola moves down. If the "b" coefficient is changed then the location of the vertex is changed. The parabola moves both horizontally and vertically.

Shape - Vertex Form:

The vertex form of a quadratic equation is f(x) = a(x - h)^2 + k.

The shape will be a parabola. If the coefficient "a" is positive then the orientation will be upward and if it is negative then the orientation will be downward.

Shape - General Form:

The general form of a quadratic equation is f(x) =ax^2 + bx + c.

The shape will be a parabola. If the coefficient "a" is positive then the orientation will be upward and if it is negative then the orientation will be downward.

The "a" coefficient cannot be zero because then there will not be an x squared term and it would then be linear.

What is the relationship between the discriminant and the amount of x-intercepts?

The discriminant is the value of b^2 - 4ac taken from the quadratic function given. If the discriminant is zero then the graph has one real root or one x-intercept. If the discriminant is positive then there are two real roots or two x-intercepts. Lastly, if the discriminant is negative then there are no real roots and no x-intercepts.

The following key vocabulary terms will be used throughout this Activity Object:

absolute value -the distance from zero on the number line

coefficient- The number placed in front of a letter representing a variable in an algebraic expression or equation, e.g., in the polynomial function f(x) = x2 + 2x + 1, the number 2 is the coefficient of x.

discriminant- for a quadratic equation, f(x) = ax2 + bx + c, the expression b2 - 4ac in the quadratic formula; this value tells you the number of zeros or x-intercepts.

horizontal shift - the amount a parabola shifts horizontally which is given by the "h" value in vertex form; f(x) = a(x - h)2 + k, it shifts the opposite amount of the number in that parenthesis.

orientation of the parabola - whether the parabola is going upward or downward (upward is a happy face and downward is a sad face).

vertical shift - the amount the parabola shifts vertically which is determined by the "k" value in the vertex form of a quadratic equation.

x-intercept - the point(s) where the parabola crosses the x-axis; it could have zero, one or two x-intercepts based on the value of the discriminant.

vertex - the point at which a parabola and its axis of symmetry intersect (maximum or minimum point).

zeros - for any function f(x) , if f(a) = 0, then "a" is a zero of the function; this in turn is giving the x-intercepts.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts Learners should be familiar with the Cartesian coordinate plane, general form of a quadratic function, quadratic function, parabola, and the vertex form of a quadratic function.
Course Algebra-2
Type of Tutorial Guided Discovery
Key Vocabulary discriminant, parabola, quadratic function